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I have to try and explain why the energy levels of an electron are difference distances apart. Here is the reasoning I've come up with so far;

We know that electrons themselves can actually transition from one energy level to another. This can be done by either supplying energy to an electron, causing it to increase an energy level $($excitation$)$ or by the electron itself emitting a photon, causing it to decrease an energy level $($relaxation$)$. We know that the latter fact is true, because when we supply energy to a gaseous substance, it illuminates, and given that we can think of visible as being discrete packets of energy, namely photons, we know this must be true. However, we know that different colours can be emitted from the same substance.

From here, we consider that $$E=hf$$and since we know that $c=f\lambda$, $\implies f=\frac{c}{\lambda}$ $$\implies E=\frac{hc}{\lambda}$$ given this inverse relationship between the wavelength of the emitted photon and its energy, we know that the certain photons emitted from a substance that emits different colours must have different energies from other photons emitted from the same substance. If we now consider the fact that the energy of a photon is measured in joules, and that the work an electron does to move from one energy level to another energy level is also measured in joules, we can look at the formula for work: $$W=Fd$$what this equation then tells us, is that there is proportionality between energy and distance, and this implies that the energy of the photon emitted by the electron is proportional to the distance the electron has to traverse.

To then answer our original question, given that different colours can be emitted from one substance, there must exist energy levels that are different distances apart.

Are there any flaws in my logic?

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You are confusing two meanings of distance.

One meaning is distance $d$ in space, as when you use $W=Fd$. The other meaning is distance (or difference $\Delta E$) in energy units.

Different colours emitted by the same atom show that there is more than one value of $\Delta E$ in an atom. They do not tell us anything about the distances in space which electrons move when they make a transition from one energy level to another.

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Your argument could conceivably make sense of different colours being due to different orbital radii (although be careful with quantum-mechanical systems, since you can't think of particles having definite orbital radii, or moving along definite paths due to work being done on them). But your title suggests you want to know why the gaps between successive orbits vary, which is a separate issue. In particular, you're asking why the $n$th orbit's radius isn't $\propto n$.

In the Bohr model - warning: it's only a toy model - the $n$th state, the first being the ground state, has an orbital radius $\propto n^2$. If you want a rough reason why, it's because solving $\dfrac{Ze^2}{4\pi\varepsilon_0 r^2}=\dfrac{mv^2}{r}$ gives $r\propto v^{-2}$ and $n\hbar = mvr \propto v^{-1}$, so $v\propto n^{-1}$ and $r\propto n^2$.

In real quantum mechanics, we get mean orbital radii; in fact, if you want nice formulae you take the mean of $r^{-1}$ instead, which will give a result $\propto n^{-2}$. And I'm also sweeping under the rug issues such as electron shielding that reduces the effective $Z$, the fact that even a classical model such as Bohr's should have elliptical orbits, and so on.

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